We develop exact Markov chain Monte Carlo methods for discretely-sampled, directly and indirectly observed diffusions. The qualification “exact” refers to the fact that the invariant and limiting distribution of the Markov chains is the exact posterior distribution of the parameters of interest. The class of processes to which our methods directly apply are those which can be simulated using the most general to date exact simulation algorithm. The article introduces various methods to boost the performance of the basic scheme, including reparametrizations and auxiliary Poisson sampling. We contrast both theoretically and empirically how this new approach compares to irreducible high frequency imputation, which is the state-of-the-art alternative for the class of processes we consider, and we uncover intriguing connections. All methods discussed are tested on typical examples.
Keywords: Exact simulation; Markov chain Monte Carlo; Stochastic differential equation; Transition density
Biography: Dr Giorgos Sermaidis is a research associate at the department of Mathematics and Statistics, Lancaster University, UK.